In [ ]:
#Today, we are going to cover recursion.
#This might be a hard concept to understand
#in the beginning. But with a little bit of
#practice,
#you'll get it.

In [4]:
def mystery(a):
if a == 0:
return 0
else:
return a + (mystery(a-1))
mystery(4)
#we can print a to see how many times
#mystery is called and what the input(a)
#is each time it is called.
#Remember printing something
#is different from returning something. We are
#printing all of these values to
#figure out what the function is doing.
#This does not change what the
#function is returning.

Out[4]:
10
In [5]:
#lets print a to see what is going on in this function
def mystery(a):
print a
if a == 0:
return 0
else:

return a + (mystery(a-1))
mystery(4)
#can you guess what this function is doing?

4
3
2
1
0

Out[5]:
10
In [ ]:
#Let us trace the function calls one by one
def mystery(a):
if a == 0:
return 0
else:
return a + (mystery(a-1))
mystery(4)
# mystery(4) = 4 + mystery(4-1) #10
# mystery(3) = 3 + mystery(3-1) #6
# mystery(2) = 2 + mystery(2-1) #3
# mystery(1) = 1 + mystery(1-1) #1
# mystery(0) = 0
#mystery(4) = 0+1+2+3+4

In [9]:
#What does this function do?
#It adds up all the numbers from 0 to a.
def mystery(a):
print a
if a == 0:
return 0
else:
return a + (mystery(a-1))
mystery(4)

4
3
2
1
0

Out[9]:
10
In [11]:
def mystery(a):
print a
return a + (mystery(a-1))
mystery(4)
#mystery(4)=4+mystery(4-1)
#mystery(3)=3+mystery(3-1)
#mystery(2)=2+mystery(2-1)
#mystery(1)=1+mystery(1-1)
#mystery(0)=0+mystery(0-1)
#mystery(-1)=-1+mystery(-1-1)
#......
#what happens now?

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---------------------------------------------------------------------------
RuntimeError                              Traceback (most recent call last)
<ipython-input-11-4368f9e77591> in <module>()
2     print a
3     return a + (mystery(a-1))
----> 4 mystery(4)
5 #mystery(4)=4+mystery(4-1)
6 #mystery(3)=3+mystery(3-1)

<ipython-input-11-4368f9e77591> in mystery(a)
1 def mystery(a):
2     print a
----> 3     return a + (mystery(a-1))
4 mystery(4)
5 #mystery(4)=4+mystery(4-1)

... last 1 frames repeated, from the frame below ...

<ipython-input-11-4368f9e77591> in mystery(a)
1 def mystery(a):
2     print a
----> 3     return a + (mystery(a-1))
4 mystery(4)
5 #mystery(4)=4+mystery(4-1)

RuntimeError: maximum recursion depth exceeded while calling a Python object
In [ ]:
#mystery keeps on calling itself forever.
# mystery(4) =
# mystery(3) =
# mystery(2) =
# mystery(1) =
# mystery(0) =
# mystery(-1) =
# mystery(-2) =
# ....

In [ ]:
#what does this function do?
#uncomment the print statemnet to see what
#a and b are everytime the function mystery2
#calles itself.
def mystery2(a,b):
#print a,b
if b == 0:
return 0
else:
return a + (mystery2(a,(b - 1)))
mystery2(5,3)
#mystery2(5,3) = 5+(mystery2(5,(3-1))) #5+5+5=15
#mystery2(5,2) = 5+(mystery2(5,(2-1))) #5+5=10
#mystery2(5,1) = 5+(mystery2(5,(1-1))) #5+0=5
#mystery2(5,0) = 0

In [ ]:
#We can implement the multiplication of 2 numbers
#a and b in as recursive function only using
#when you multiply 5*3, what do you do?
#5*3 is 5+5+5. The function below adds 5 to itself
#3 times. In genera, the function below takes
#inputs a and b and adds a to itself b times.
#This is the same as performing a*b.
def mystery2(a,b):
print a,b
if b == 0:
return 0
else:
return a + (mystery2(a,(b - 1)))
mystery2(5,3)

In [ ]:
#Lets see another an example.
#consider the Fibonacci sequence
#1, 1, 2, 3, 5, 8, 13, . . ..
#This sequence is defined by the 0th and 1st
#Fibonacci numbers both being 1,
#and subsequent Fibonacci numbers being
#the sum of the previous two.

#The Fibonacci sequence is very well known in
#Math and
#appears frequently in nature.
#It also has applications
#in various computer science algorithms.

#We will not discuss the Fibonacci sequence
#in detail
#in this class but if you are interested in
#learning
#https://en.wikipedia.org/wiki/Fibonacci_number

In [ ]:
#The Fibonacci sequence is defined by the 0th
#and 1st
#Fibonacci numbers both being 1,
#and subsequent Fibonacci numbers being
#the sum of the previous two
#F(i) is 1 if i=0 or i=1
#
#F(i) is F(i−1) + F(i−2) otherwise

#So what is F(0)?
#    1 because i is 1
#What is F(1)?
#   1 because i is 1
#What is F(2)
#   It is F(i-1) + F(i-2) #F(1)+F(0)
#   which is 1+1=2
#F(3) is F(i-1)+F(i-2)=F(3-1)+F(3-2)
#   which is F(2)+F(1)=2+1=3
#F(4) is F(4-1)+F(4-2)=F(3)+F(2)=3+2=5
# ....And it continues this way

In [ ]:
##Now let us make a function that takes i and
#returns the ith Fibonacci number.
#We can do it the way we have learned how to.
#First we write in English how we want to create
#the function.

In [12]:
#Now let us make a function that takes i and
#returns the ith Fibonacci number.
#We can do it the way we have learned how to.
#lets call the function fibonacci
def fibonacci(i):
fib_i=1 #it is called initialization.
fib_i_prev=1
fib_i_prev_prev=1

if i==0 or i==1:
return fib_i
for x in range(2,i+1): #[2,3,4....i]
fib_i = fib_i_prev + fib_i_prev_prev
fib_i_prev_prev=fib_i_prev
fib_i_prev=fib_i
return fib_i
fibonacci(5)

Out[12]:
8
In [13]:
#give me a function fibonacci. It takes an input i,
#and returns the ith fibonacci number.
#Remember the formula.
#if i=0, or i=1, what is the fibonacci number? 1.
#ohterwise what is it fibonacci(i)=fibonacci(i-1)+
#fibonacci(i-2)
#fibonacci(0) #1
#fibonacci(1) #1
#fibonacci(2) #2 fibonacci(1)+fibonacci(0)
def fibonacci(i):
if i==0 or i==1:
return 1
else:
return fibonacci(i-1)+fibonacci(i-2)
fibonacci(5)
#fibonacci(5)=fibonacci(4)+fibonacci(3) #8
#fibonacci(4)=fibonacci(3)+fibonacci(2) #5
#fibonacci(3)=fibonacci(2)+fibonacci(1) #3
#fibonacci(2)=fibonacci(1)+fibonacci(0) #2
#fibonacci(1)=1

Out[13]:
8
In [ ]:
#Although we could implement the Fibonacci sequence
#in a very complicated way without using recursion,
#the following function using recursion
#is very simple and easier to understand
def fibonacci(i):
if i<2:
return 1
return fibonacci(i-1) + fibonacci(i-2)
fibonacci(5)

#compare this function to the function we
#wrote above
#which is more complicated and hard to understand